The edge spectrum of the saturation number for small paths

Abstract

Let H be a simple graph. A graph G is called an H-saturated graph if H is not a subgraph of G, but adding any missing edge to G will produce a copy of H. Denote by SAT(n,H) the set of all H-saturated graphs G with order n. Then the saturation number sat(n,H) is defined as minG∈SAT(n,H)|E(G)|, and the extremal number ex(n,H) is defined as maxG∈SAT(n,H)|E(G)|. A natural question is that of whether we can find an H-saturated graph with m edges for any sat(n,H)≤m≤ex(n,H). The set of all possible values m is called the edge spectrum for H-saturated graphs. In this paper we investigate the edge spectrum for Pi-saturated graphs, where 2≤i≤6. It is trivial for the case of P2 that the saturated graph must be an empty graph.